Consider for example a census or sample survey of incomes. Even though each individual income may follow Geometric Brownian Motion, the time during which it has been so evolving will vary from individual to individual. If recruitment to the workforce has been growing at a more or less constant rate, the distribution of time in the workforce of any individual will follow an exponential distribution (numeraire one say), the current distribution of incomes should be that of a GBM observed after an exponentially distributed time T. This distribution is what I call a double Pareto distribution, with a density proportional to x-a-1 for x>1 and proportional to xb-1 for x<1 (for details and proof see the Appendix). More realistically individual starting incomes will also vary and evolve over time, say as another GBM. In this case current incomes can be represented as the product of a double Pareto random variable with an independent lognormal random variable, and will exhibit power law behaviour in both tails (for details see Reed, 2000 a). Thus, not only does this simple model offer a plausible explanation of the Pareto Law of Incomes (upper tail), it also predicts power-law behaviour in the lower tail. Does such behaviour occur? The figure below (left-hand
panels, both axes logarithmic) for data on all U.S. male income earners in 1998 (U.S. Census Bureau, 1999) suggests that power law behaviour occurs in both tails. Indeed lower-tail powerlaw behaviour has been identified before (Champernowne, 1953) but is not apparently widely recognized.